I have three homework problems that I got stuck on while, trying to solve. I'd really appreciate it if anyone could help me with these. Here they are

For # 14 & 16 use a rectangular coordinate system to plot u= [vertical: 5 2]
, v=[vert: -2 4], and their images under the given transformation T. (Make a large sketch for each exercise.) Describe geometrically what T does to each vector x in R^2.

I have three homework problems that I got stuck on while, trying to solve. I'd really appreciate it if anyone could help me with these. Here they are

For # 14 & 16 use a rectangular coordinate system to plot u= [vertical: 5 2]
, v=[vert: -2 4], and their images under the given transformation T. (Make a large sketch for each exercise.) Describe geometrically what T does to each vector x in R^2.

#34 Let T: R^n----> R^m be a linear transformation. Show that if T maps two linearly independent vectors onto a linearly dependent set, then the equation T(x)=0 has a notrivial solution.

Suppose that {u,v} are linearly independent in R^n and we have T(u) and T(v) are linearly dependent. There exists weights c_1, c_2 (both of them not zero), such that:

c_1*T(u) + c_2*T(v) = 0

Since T is linear, T(c_1*u + c_2*v) = 0. Thus,

x= c_1*u+ c_2*v will satisfy T(x) = 0. And, we know that it's impossible for x to be the zero vector since that'd imply that a nontrivial linear combination of u and v is zero, and we know that is impossible since u and v are linearly independent.

Therefore, we can conlude that the equation T(x) = 0 has a nontrivial solution.